Lithographic apparatus, method of determining a model parameter, device manufacturing method, and device manufactured thereby

ABSTRACT

A method according to one embodiment of the invention relates to determining at least one parameter of a model that provides information about a position of an object. The object has a plurality of alignment marks of which desired positions are known. The method includes measuring a plurality of positional parameters for each alignment mark. Based on the measured plurality of positional parameters, which are weighted with weighing coefficients, at least one parameter of the model of the object is determined. The numerical value of each weighing coefficient is determined together with the at least one parameter of the model.

FIELD OF THE INVENTION

The present invention relates to positioning of an object and tolithographic projection apparatus and methods.

BACKGROUND

The term “patterning structure” as here employed should be broadlyinterpreted as referring to any structure or field that may be used toendow an incoming radiation beam with a patterned cross-section,corresponding to a pattern that is to be created in a target portion ofa substrate; the term “light valve” can also be used in this context. Itshould be appreciated that the pattern “displayed” on the patterningstructure may differ substantially from the pattern eventuallytransferred to e.g. a substrate or layer thereof (e.g. where pre-biasingof features, optical proximity correction features, phase and/orpolarization variation techniques, and/or multiple exposure techniquesare used). Generally, such a pattern will correspond to a particularfunctional layer in a device being created in the target portion, suchas an integrated circuit or other device (see below). Patterningstructure may be reflective and/or transmissive. Examples of patterningstructure include:

A mask. The concept of a mask is well known in lithography, and itincludes mask types such as binary, alternating phase-shift, andattenuated phase-shift, as well as various hybrid mask types. Placementof such a mask in the radiation beam causes selective transmission (inthe case of a transmissive mask) or reflection (in the case of areflective mask) of the radiation impinging on the mask, according tothe pattern on the mask. In the case of a mask, the support structurewill generally be a mask table, which ensures that the mask can be heldat a desired position in the incoming radiation beam, and that it can bemoved relative to the beam if so desired.

A programmable mirror array. One example of such a device is amatrix-addressable surface having a viscoelastic control layer and areflective surface. The basic principle behind such an apparatus is that(for example) addressed areas of the reflective surface reflect incidentlight as diffracted light, whereas unaddressed areas reflect incidentlight as undiffracted light. Using an appropriate filter, theundiffracted light can be filtered out of the reflected beam, leavingonly the diffracted light behind; in this manner, the beam becomespatterned according to the addressing pattern of the matrix-addressablesurface. An array of grating light valves (GLVs) may also be used in acorresponding manner, where each GLV may include a plurality ofreflective ribbons that can be deformed relative to one another (e.g. byapplication of an electric potential) to form a grating that reflectsincident light as diffracted light. A further alternative embodiment ofa programmable mirror array employs a matrix arrangement of very small(possibly microscopic) mirrors, each of which can be individually tiltedabout an axis by applying a suitable localized electric field, or byemploying piezoelectric actuation means. For example, the mirrors may bematrix-addressable, such that addressed mirrors will reflect an incomingradiation beam in a different direction to unaddressed mirrors; in thismanner, the reflected beam is patterned according to the addressingpattern of the matrix-addressable mirrors. The required matrixaddressing can be performed using suitable electronic means. In both ofthe situations described hereabove, the patterning structure cancomprise one or more programmable mirror arrays. More information onmirror arrays as here referred to can be gleaned, for example, from U.S.Pat. Nos. 5,296,891 and 5,523,193 and PCT patent applications WO98/38597 and WO 98/33096, which documents are incorporated herein byreference. In the case of a programmable mirror array, the supportstructure may be embodied as a frame or table, for example, which may befixed or movable as required.

A programmable LCD panel. An example of such a construction is given inU.S. Pat. No. 5,229,872, which is incorporated herein by reference. Asabove, the support structure in this case may be embodied as a frame ortable, for example, which may be fixed or movable as required.

For purposes of simplicity, the rest of this text may, at certainlocations, specifically direct itself to examples involving a mask (or“reticle”) and mask table (or “reticle table”); however, the generalprinciples discussed in such instances should be seen in the broadercontext of the patterning structure as hereabove set forth.

A lithographic apparatus may be used to apply a desired pattern onto asurface (e.g. a target portion of a substrate). Lithographic projectionapparatus can be used, for example, in the manufacture of integratedcircuits (ICs). In such a case, the patterning structure may generate acircuit pattern corresponding to an individual layer of the IC, and thispattern can be imaged onto a target portion (e.g. comprising one or moredies and/or portion(s) thereof) on a substrate (e.g. a wafer of siliconor other semiconductor material) that has been coated with a layer ofradiation-sensitive material (e.g. resist). In general, a single waferwill contain a whole matrix or network of adjacent target portions thatare successively irradiated via the projection system (e.g. one at atime).

Among current apparatus that employ patterning by a mask on a masktable, a distinction can be made between two different types of machine.In one type of lithographic projection apparatus, each target portion isirradiated by exposing the entire mask pattern onto the target portionat once; such an apparatus is commonly referred to as a wafer stepper.In an alternative apparatus—commonly referred to as a step-and-scanapparatus—each target portion is irradiated by progressively scanningthe mask pattern under the projection beam in a given referencedirection (the “scanning” direction) while synchronously scanning thesubstrate table parallel or anti-parallel to this direction; since, ingeneral, the projection system will have a magnification factor M(generally <1), the speed V at which the substrate table is scanned willbe a factor M times that at which the mask table is scanned. Aprojection beam in a scanning type of apparatus may have the form of aslit with a slit width in the scanning direction. More information withregard to lithographic devices as here described can be gleaned, forexample, from U.S. Pat. No. 6,046,792, which is incorporated herein byreference.

In a manufacturing process using a lithographic projection apparatus, apattern (e.g. in a mask) is imaged onto a substrate that is at leastpartially covered by a layer of radiation-sensitive material (e.g.resist). Prior to this imaging procedure, the substrate may undergovarious other procedures such as priming, resist coating, and/or a softbake. After exposure, the substrate may be subjected to other proceduressuch as a post-exposure bake (PEB), development, a hard bake, and/ormeasurement/inspection of the imaged features. This set of proceduresmay be used as a basis to pattern an individual layer of a device (e.g.an IC). For example, these transfer procedures may result in a patternedlayer of resist on the substrate. One or more pattern processes mayfollow, such as deposition, etching, ion-implantation (doping),metallization, oxidation, chemo-mechanical polishing, etc., all of whichmay be intended to create, modify, or finish an individual layer. Ifseveral layers are required, then the whole procedure, or a variantthereof, may be repeated for each new layer. Eventually, an array ofdevices will be present on the substrate (wafer). These devices are thenseparated from one another by a technique such as dicing or sawing,whence the individual devices can be mounted on a carrier, connected topins, etc. Further information regarding such processes can be obtained,for example, from the book “Microchip Fabrication: A Practical Guide toSemiconductor Processing,” Third Edition, by Peter van Zant, McGraw HillPublishing Co., 1997, ISBN 0-07-067250-4.

A substrate as referred to herein may be processed before or afterexposure: for example, in a track (a tool that typically applies a layerof resist to a substrate and develops the exposed resist) or a metrologyor inspection tool. Where applicable, the disclosure herein may beapplied to such and other substrate processing tools. Further, thesubstrate may be processed more than once (for example, in order tocreate a multi-layer IC), so that the term substrate as used herein mayalso refer to a substrate that already contains multiple processedlayers.

The term “projection system” should be broadly interpreted asencompassing various types of projection system, including refractiveoptics, reflective optics, and catadioptric systems, for example. Aparticular projection system may be selected based on factors such as atype of exposure radiation used, any immersion fluid(s) or gas-filledareas in the exposure path, whether a vacuum is used in all or part ofthe exposure path, etc. For the sake of simplicity, the projectionsystem may hereinafter be referred to as the “lens.” The radiationsystem may also include components operating according to any of thesedesign types for directing, shaping, reducing, enlarging, patterning,and/or otherwise controlling the projection beam of radiation, and suchcomponents may also be referred to below, collectively or singularly, asa “lens.”

Further, the lithographic apparatus may be of a type having two or moresubstrate tables (and/or two or more mask tables). In such “multiplestage” devices the additional tables may be used in parallel, orpreparatory steps may be carried out on one or more tables while one ormore other tables are being used for exposures. Dual stage lithographicapparatus are described, for example, in U.S. Pat. No. 5,969,441 and PCTApplication No. WO 98/40791, which documents are incorporated herein byreference.

The lithographic apparatus may also be of a type wherein the substrateis immersed in a liquid having a relatively high refractive index (e.g.water) so as to fill a space between the final element of the projectionsystem and the substrate. Immersion liquids may also be applied to otherspaces in the lithographic apparatus, for example, between the mask andthe first element of the projection system. The use of immersiontechniques to increase the effective numerical aperture of projectionsystems is well known in the art.

In the present document, the terms “radiation” and “beam” are used toencompass all types of electromagnetic radiation, including ultravioletradiation (e.g. with a wavelength of 365, 248, 193, 157 or 126 nm) andEUV (extreme ultra-violet radiation, e.g. having a wavelength in therange 5-20 nm), as well as particle beams (such as ion or electronbeams).

Although specific reference may be made in this text to the use oflithographic apparatus in the manufacture of ICs, it should beexplicitly understood that such an apparatus has many other possibleapplications. For example, it may be employed in the manufacture ofintegrated optical systems, guidance and detection patterns for magneticdomain memories, liquid-crystal display panels, thin-film magneticheads, DNA analysis devices, etc. The skilled artisan will appreciatethat, in the context of such alternative applications, any use of theterms “wafer” or “die” in this text should be considered as beingreplaced by the more general terms “substrate” and “target portion,”respectively.

As described above, a number of patterned layers may be created on asubstrate. In order to create an operating device or to provide optimalperformance, it may be desirable or even necessary for the patterns oflayers positioned on top of each other to be well aligned with respectto each other. Such a condition may be accomplished by accuratelypositioning the substrate with respect to the mask and the projectionbeam.

In the first place, it may be desirable or necessary for the substrateto be in the focal plane of the patterned beam, in order to obtain asharp image of the patterning structure (a process also known as“leveling”). The direction associated with this distance is called thez-direction.

Secondly, it may be desirable or necessary to accurately set theposition of the substrate in the directions perpendicular to thez-direction, i.e. the x- and y-direction, in order to position thedifferent layers correctly on top of each other (a process also known as“aligning”). Accurate aligning is generally done by accuratelydetermining the position of the substrate relative to a substrate tablewhich holds the substrate and determining the position of the substratetable with respect to the mask and projection beam. Alignment may bedone using an alignment system, as described for instance in U.S. Pat.No. 6,297,876, which document is incorporated herein by reference.

The shape of the substrate may differ from an ideal shape of thesubstrate. Differences of the shape may be caused by the shape of theunderlying surface (for instance, a pimple structure of a substratetable) but may also be influenced by a clamp used to clamp the substrateto, for instance, the substrate table. For example, the forces generatedby the clamp may deform the substrate, at least locally. In order toproject a patterned beam as accurately as possible, information aboutthe exact shape of the substrate may be required.

Information about the position and/or shape of the substrate may beobtained by measuring the position of one or more alignment marksprovided on the substrate. Alignment marks may be arranged to diffractlight when illuminated, such that the diffracted light may be detectedby one or more sensors. From the detected signal, information may bederived about the position of the mark. For instance, such alignmentmarks may be formed by gratings that produce a diffraction pattern whenilluminated with an alignment beam. Measuring the position of adiffraction order of a diffraction pattern, as produced by the alignmentmark with respect to the sensor(s), may be used to provide informationabout the position of the alignment mark and thus the position of thesubstrate.

However, the results of known methods are not always accurate enough.Therefore, it is desirable to obtain a method that is more accurate.

SUMMARY

A method according to one embodiment of the invention includesdetermining at least one parameter of a model. The model providesinformation about a position of an object that is provided with aplurality of alignment marks of which desired positions are known. Themethod includes measuring a plurality of positional parameters for eachalignment mark. Based on the measured plurality of positionalparameters, the at least one parameter of the model of the object isdetermined. The plurality of positional parameters for each alignmentmark are weighted with weighing coefficients, and the numerical value ofat least one of the weighing coefficient is determined together with theat least one parameter of the model. Modifications of such a method aredisclosed, including methods for manufacturing devices, as well asapparatus that may be used to determine a position of an object.

BRIEF DESCRIPTION OF THE DRAWINGS

Embodiments of the invention will now be described, by way of exampleonly, with reference to the accompanying schematic drawings in which:

FIG. 1 depicts a lithographic apparatus according to an embodiment ofthe invention.

FIG. 2 depicts a measurement setup including a substrate W according toan embodiment of the invention.

In the Figures, corresponding reference symbols indicate correspondingparts.

DETAILED DESCRIPTION

Embodiments of the invention include, for example, a method ofdetermining at least one parameter of a model that provides informationabout a position of an object, such as a substrate (and possibly otherinformation, such as information pertaining to deformation of theobject).

FIG. 1 schematically depicts a lithographic projection apparatusaccording to a particular embodiment of the invention. The apparatuscomprises:

A radiation system configured to supply (e.g. having structure capableof supplying) a projection beam of radiation (e.g. UV or EUV radiation).In this particular example, the radiation system RS comprises aradiation source SO, a beam delivery system BD, and an illuminationsystem including adjusting structure AM (e.g. for setting anillumination node), an integrator IN, and condensing optics CO;

A support structure configured to support a patterning structure capableof patterning the projection beam. In this example, a first object table(mask table) MT is provided with a mask holder for holding a mask MA(e.g. a reticle), and is connected to a first positioning structure PMfor accurately positioning the mask with respect to item PL;

A second object table (substrate table) configured to hold a substrate.In this example, substrate table WT is provided with a substrate holderfor holding a substrate W (e.g. a resist-coated semiconductor wafer),and is connected to a second positioning structure PW for accuratelypositioning the substrate with respect to item PL and (e.g.interferometric) measurement structure IF, which is configured toaccurately indicate the position of the substrate and/or substrate tablewith respect to lens PL; and

A projection system (“lens”) configured to project the patterned beam.In this example, projection system PL (e.g. a refractive lens group, acatadioptric or catoptric system, and/or a mirror system) is configuredto image an irradiated portion of the mask MA onto a target portion C(e.g. comprising one or more dies and/or portion(s) thereof) of thesubstrate W. Alternatively, the projection system may project images ofsecondary sources for which the elements of a programmable patterningstructure may act as shutters. The projection system may also include amicrolens array (MLA), e.g. to form the secondary sources and to projectmicrospots onto the substrate.

As here depicted, the apparatus is of a transmissive type (e.g. has atransmissive mask). However, in general, it may also be of a reflectivetype, for example (e.g. with a reflective mask). Alternatively, theapparatus may employ another kind of patterning structure, such as aprogrammable mirror array of a type as referred to above.

The source SO (e.g. a mercury lamp, an excimer laser, an electron gun, alaser-produced plasma source or discharge plasma source, or an undulatorprovided around the path of an electron beam in a storage ring orsynchrotron) produces a beam of radiation. This beam is fed into anillumination system (illuminator) IL, either directly or after havingtraversed a beam delivery system BD, which may include suitabledirecting mirrors and/or a conditioning structure or field, such as abeam expander. The illuminator IL may comprise an adjusting structure orfield AM for setting the outer and/or inner radial extent (commonlyreferred to as σ-outer and σ-inner, respectively) of the intensitydistribution in a pupil plane of the illuminator, which may affect theangular distribution of the radiation energy delivered by the projectionbeam at, for example, the substrate. In addition, the apparatus willgenerally comprise various other components, such as an integrator INand a condenser CO. In this way, the beam PB impinging on the mask MAmay have a desired uniformity and intensity distribution in itscross-section.

It should be noted with regard to FIG. 1 that the source LA may bewithin the housing of the lithographic projection apparatus (as is oftenthe case when the source LA is a mercury lamp, for example), but that itmay also be remote from the lithographic projection apparatus, theradiation beam which it produces being led into the apparatus (e.g. withthe aid of suitable direction mirrors); this latter scenario is oftenthe case when the source LA is an excimer laser. The current inventionand claims encompass both of these scenarios.

The beam PB subsequently intercepts the mask MA, which is held on a masktable MT. Having traversed (alternatively, having been selectivelyreflected by) the mask MA, the beam PB passes through the lens PL, whichfocuses the beam PB onto a target portion C of the substrate W. With theaid of the second positioning structure PW (and interferometricmeasuring structure IF), the substrate table WT can be moved accurately,e.g. so as to position different target portions C in the path of thebeam PB. Similarly, the first positioning structure PM can be used toaccurately position the mask MA with respect to the path of the beam PB,e.g. after mechanical retrieval of the mask MA from a mask library, orduring a scan. In general, movement of the object tables MT, WT will berealized with the aid of a long-stroke module (coarse positioning) and ashort-stroke module (fine positioning), which are not explicitlydepicted in FIG. 1. However, in the case of a wafer stepper (as opposedto a step-and-scan apparatus) the mask table MT may just be connected toa short stroke actuator, or may be fixed. Mask MA and substrate W may bealigned using mask alignment marks M1, M2 and substrate alignment marksP1, P2.

The depicted apparatus can be used in several different modes:

1. In step mode, the mask table MT is kept essentially stationary, andan entire mask image is projected at once (i.e. in a single “flash”)onto a target portion C. The substrate table WT is then shifted in the xand/or y directions so that a different target portion C can beirradiated by the beam PB;

2. In scan mode, essentially the same scenario applies, except that agiven target portion C is not exposed in a single “flash.” Instead, themask table MT is movable in a given direction (the so-called “scandirection”, e.g. the y direction) with a speed v, so that the projectionbeam PB is caused to scan over a mask image. Concurrently, the substratetable WT is simultaneously moved in the same or opposite direction at aspeed V=Mv, in which M is the magnification of the lens PL (typically,M=¼ or ⅕). In this manner, a relatively large target portion C can beexposed, without having to compromise on resolution.

3. In another mode, the mask table MT is kept essentially stationaryholding a programmable patterning structure, and the substrate table WTis moved or scanned while a pattern imparted to the projection beam isprojected onto a target portion C. In this mode, generally a pulsedradiation source is employed and the programmable patterning structureis updated as required after each movement of the substrate table WT orin between successive radiation pulses during a scan. This mode ofoperation can be readily applied to maskless lithography that utilizesprogrammable patterning structure, such as a programmable mirror arrayof a type as referred to above.

Combinations and/or variations on the above-described modes of use orentirely different modes of use may also be employed.

A diffraction pattern (e.g. as generated by an alignment beam projectedto an alignment mark) may comprise a number of diffraction orders, andsome number (for instance, seven) of the diffraction orders may bemeasured. Each diffraction orders line comprises positional informationabout the alignment mark. In many cases, a position of the alignmentmark can be determined based on the determined position of a singlediffraction order, but more accurate results may be obtained when morediffraction orders are taken into account.

It is known to calculate a wafer model (i.e. numerical values for thetranslation T, the rotation R and/or the expansion Exp of a substrate)before exposure, that can be used to compute the position, expansionand/or orientation of a substrate based on the measured positions of thediffraction orders. Such a technique may include assigning a weighingcoefficient to each diffraction order. If, for instance, sevendiffraction orders (i=1, 2, . . . , 7) are measured for the x-direction,7 weighing coefficients α_(i) may be defined. These weighingcoefficients may be positive or negative, but it may be desirable forthe weighing coefficients to be defined and/or normalized such thattheir sum equals one: ${\sum\limits_{i}^{\quad}\quad\alpha_{i}} = 1.$

For the y-direction, other weighing coefficients β_(i) may be defined.Alignment beams may be used that comprise more than one color (forinstance, red and green), so that more diffraction orders may begenerated by the alignment beams, and the wafer model may comprise moreweighing coefficients.

A single alignment mark may comprise more than one grating. For example,a first grating might be directed in a first direction (x), while asecond grating might be directed in a second direction (y), which may besubstantially perpendicular to the first direction. In such cases, asingle alignment mark may be used to obtain information on position inmore than one direction or dimension.

Usually the position of a substrate (or portion thereof) will bedetermined based on positional information derived from a number ofalignment marks, which may be spread over the surface of the substrate.In general, the more alignment marks are measured, the more accuratelythe position of the substrate may be determined.

After the position of a plurality of alignment marks across the surfaceof the substrate is determined, this information may be supplied to adeformation model. For example, the deformation model may be arranged todetermine the position and orientation of each target portion of thesubstrate. The deformation model may also be arranged to determinedeformation within a target portion. The results of such a deformationmodel may be used to make sure that each target portion will becorrectly aligned during exposure.

The position of the substrate may be expressed as a translation T, arotation R, and an expansion Exp. The translation may be in thex-direction T_(x) and/or in the y-direction T_(y). The rotation may be arotation of the x-axis around the z-axis R_(zx) and/or a rotation of they-axis around the z-axis R_(zy). The expansion may be an expansion inthe direction of the x-axis Exp_(x) and/or in the direction of they-axis Exp_(y).

With the measurements of the diffraction orders of the different marks,together with the weighing coefficients, a wafer model comprising thetranslation, rotation, and expansion may be determined. Such a model maybe determined using, e.g., a least square method, as will be understoodby a person skilled in the art and as will be explained below. Computingthe wafer model parameters may include minimizing the differencesbetween the computed positions of the alignment marks and the measuredpositions of the alignment marks.

In a method according to an embodiment of the invention, the numericalvalue of each weighing coefficient is determined together with the atleast one parameter of the model. Such a method provides a dynamicrecipe, in that it may adjust its weighing parameters to each newsituation. Therefore, it is flexible and may provide results that aremore accurate. Such a method can be applied for determining theparameters of a model, such as the translation T, the rotation R, andthe expansion Exp. From these, the position and/or deformation of anobject, such as a substrate, may be determined in a single direction,for instance, the x-direction, but may also very well be used todetermine the position and/or deformation of an object in twodirections, the x and y direction, and may also be applied to determinethe position and/or deformation of the object in three directions, x, yand z direction. Furthermore, the method might be used to determine therotational position of the object with respect to one or more of thesedirections. Thus, one potential advantage of such a method is increasedaccuracy in position determination.

In a method according to a further embodiment of the invention, the atleast one parameter of the model is at least one of translation,rotation and expansion. With these three parameters, the position anddeformation may very well be expressed. The translation may be atranslation in a first direction and/or a translation in a seconddirection. The rotation may be a rotation of the object as a wholearound a certain axis, but may also express rotational deformation, forexample, as a rotational position of a first axis with respect to asecond axis (e.g. the x axis with respect to the y axis). Also theexpansion may express lateral deformation (e.g. different expansions fordifferent directions).

In a substrate according to a further embodiment of the invention, theplurality of alignment marks are formed as diffractive elements (such asmultigratings), such that the plurality of positional parameters may bedetermined, e.g., by diffraction lines generated by projecting analignment beam to the plurality of alignment marks. Diffractiveelements, such as gratings, are well suited for providing positionalinformation. A multigrating mark may have three good signal orders foreach color. Information about the position of such a grating can easilybe obtained by projecting an alignment beam to the diffractive elementand measuring the position of the diffractive pattern.

In a method according to a further embodiment of the invention, the atleast one parameter of the model is solved by minimizing the expression$\sum\limits_{\forall i}^{\quad}\quad\sqrt{\left( {x_{{meas},i} - x_{nom}} \right)^{2} + \left( {y_{{meas},i} - y_{nom}} \right)^{2}}$

for a plurality of (possibly all of) the alignment marks, wherex_(meas,i) and y_(meas,i) denote measured positions of the alignmentmarks based on an i-th positional parameter in an x-direction andy-direction, respectively, and x_(nom) and y_(nom) denote desiredpositions in the x and y-direction, respectively. Such a technique maybe used as a relatively easy way of computing wafer model parameters.

A method according to a further embodiment of the invention includessetting the value of a weighing coefficient to zero when the signalstrength of a corresponding positional parameter is below a certainthreshold. If the measured signal of, for instance, a particulardiffraction order is too weak, the signal could be ignored by settingthe corresponding weighing coefficient to zero.

In a method according to a further embodiment of the invention, theobject is a substrate. In the case of a substrate processed by alithographic apparatus, it may be necessary to determine the positionand shape of a substrate accurately in order to allow accurateprojection of a patterned beam.

A device manufacturing method according to another embodiment of theinvention includes providing a substrate, providing a beam of radiationusing an illumination system, using a patterning structure to impart theprojection beam with a pattern in its cross-section, and projecting thepatterned beam of radiation onto a target portion of the substrate,wherein a method as described herein is performed before the patternedbeam is projected.

In a method according to a further embodiment of the invention, thenumerical value of each weighing coefficient is determined based onmeasurements of at least one substrate, and the determined numericalvalue of each weighing coefficient is used during determination of theat least one parameter of the model for subsequent substrates. Onepotential advantage of such a method is that a calculation procedure asdescribed herein only needs to be done a relatively low number of times,for instance, only for the first 3 substrates from a batch. In at leastsome circumstances, it may be appropriate to assume that the sameweighing coefficients can be used for subsequent substrates from thatbatch.

A lithographic apparatus according to a further embodiment of theinvention includes an illumination system for providing a beam ofradiation; a support structure for supporting a patterning structure,the patterning structure serving to impart the beam with a pattern inits cross-section; a substrate table for holding a substrate; and aprojection system for projecting the patterned beam onto a targetportion of the substrate. Such a lithographic apparatus also includes aprocessing unit, a beam generator and a sensor. The beam generator (e.g.a laser emitter) is arranged to project an alignment beam to a pluralityof alignment marks formed on the substrate, generating at least twopositional parameters for each alignment mark, of which desiredpositions are known. The sensor is arranged to measure the positionalparameters and transfer the measured positional parameters to theprocessing unit. The processing unit, which is arranged to communicatewith the sensor, is also arranged to determine at least one parameter ofa model, based on the measured plurality of positional parameters, whilethe plurality of positional parameters for each alignment mark areweighted with weighing coefficients, the model providing informationabout at least a position of the substrate. In such an apparatus, theprocessing unit determines the numerical value of each weighingcoefficient together with the at least one parameter of the model.

In a method according to a further embodiment of the invention, theweighing coefficients α_(i) are not determined beforehand (e.g. as in astatic recipe), but are variables that can be varied while solving theleast squares solution in order to determine the wafer model (i.e. adynamic recipe).

As discussed above, the position and/or shape of the substrate W may beexpressed as a translation (T_(x), T_(y)), a rotation of the x-axisaround the z-axis (R_(zx), here denoted as R_(x)) and a rotation of they-axis around the z-axis (R_(zy), here denoted as R_(y)) and anexpansion in the x-direction (Exp_(x)) and the y-direction (Exp_(y)).One such model is called the 6-parameter wafer model with, for theX-direction:T _(x) +EXP _(x) x _(nom)(X _(N))−R _(x) y _(nom)(X _(N))=x _(meas)(X_(N))−x _(nom)(X_(N))   (1)and for the Y direction:

T _(y) +EXP _(y) y _(nom)(Y _(N))+R _(y) x _(nom)(Y _(N))=y _(meas)(Y_(N))−y _(nom)(Y _(N)).

Here, X_(N) and Y_(N) denote the X and Y values obtained by measuringthe Nth alignment mark, x_(meas) and y_(meas) denote the measured valuesof X and Y, and x_(nom) and y_(nom) denote a desired location of themeasured alignment mark.

For reasons of simplicity, in the embodiment discussed below, R_(zx)will be assumed to equal R_(zy) (i.e. no rotational deformation of thesubstrate W) and Exp_(x) will be assumed to equal Exp_(y) (i.e. theexpansion is equal in the x and y direction). One such model is calledthe 4-parameter wafer model (T_(x), T_(y), Exp, R), although it must beunderstood that embodiments of the invention may be applied as well towafer models having six parameters, and also to models having othercombinations of parameters.

As is depicted in FIG. 2, a number of alignment marks 10 are provided ona substrate W. The alignment marks 10 are arranged to generate a numberof positional indicators, providing information about the position ofeach alignment mark 10 to a sensor 11. The sensor 11 is arranged toprovide its measurements to a processing unit 13. The processing unit 13may be arranged to store these measurements in a memory unit 14. Theremay be applied a plurality of sensors instead of one. Also, thesubstrate may have other alignment marks that do not provide informationto sensor 11 in such a manner and/or whose information is not includedin one or more calculations as described herein.

In FIG. 2, the alignment mark 10 is a diffractive element, for instance,formed by a grating. A beam generator 12 is arranged to provide analignment beam AB (e.g. a laser beam) which is projected to one of thealignment marks 10. The beam generator 12 may be controlled by theprocessing unit 13. The alignment beam AB in combination with alignmentmark 10, here formed as a grating, generates a number of diffractionorders of which the position is measured by the sensor 11. Themeasurements of the sensor 11 are transferred to the processing unit 13,that is arranged to compute the position of the substrate W according tothe method as described below. The results may be stored in the memoryunit 14.

In order to obtain positional information about the different alignmentmarks 10, the alignment beam AB needs to be projected to these otheralignment marks 10. This may be done by moving the substrate W withrespect to the beam generator 12 and the sensor 11. Therefore, thesubstrate W is preferably positioned on an implementation of substratetable WT that is moveable (e.g. via a positioning structure W). However,it is also possible to move the beam, the beam generator 12, and/or thesensor with respect to the substrate W.

In general, the grid parameters T_(x), T_(y), R and Exp in case of Nalignment marks 10 are obtained by solving a set of equations using amethod of least squares. All alignment marks 10 each comprise a gratingdirected in the X- and Y-direction, so N pairs (X₁, X₂, . . . X_(N); Y₁,Y₂, . . . Y_(N)) are obtained by least-squares solving a set of 2Nequations, consisting of N equations for X: $\begin{matrix}\begin{matrix}\begin{matrix}{{T_{x} + {{Exp}\quad{x_{nom}\left( X_{1} \right)}} - {R\quad{y_{nom}\left( X_{1} \right)}}} = {{x_{meas}\left( X_{1} \right)} - {x_{nom}\left( X_{1} \right)}}} \\{{T_{x} + {{Exp}\quad{x_{nom}\left( X_{2} \right)}} - {R\quad{y_{nom}\left( X_{2} \right)}}} = {{x_{meas}\left( X_{2} \right)} - {x_{nom}\left( X_{2} \right)}}}\end{matrix} \\\vdots \\{{T_{x} + {{Exp}\quad{x_{nom}\left( X_{N} \right)}} - {R\quad{y_{nom}\left( X_{N} \right)}}} = {{x_{meas}\left( X_{N} \right)} - {x_{nom}\left( X_{N} \right)}}}\end{matrix} & (2)\end{matrix}$and N equations for Y: $\begin{matrix}{{T_{y} + {{Exp}\quad{y_{nom}\left( Y_{1} \right)}} + {R\quad{x_{nom}\left( Y_{1} \right)}}} = {{y_{meas}\left( Y_{1} \right)} - {y_{nom}\left( Y_{1} \right)}}} \\{{T_{y} + {{Exp}\quad{y_{nom}\left( Y_{2} \right)}} + {R\quad{x_{nom}\left( Y_{2} \right)}}} = {{y_{meas}\left( Y_{2} \right)} - {y_{nom}\left( Y_{2} \right)}}} \\\vdots \\{{T_{y} + {{Exp}\quad{y_{nom}\left( Y_{N} \right)}} + {R\quad{x_{nom}\left( Y_{N} \right)}}} = {{y_{meas}\left( Y_{N} \right)} - {y_{nom}\left( Y_{N} \right)}}}\end{matrix}$

In a method according to some embodiments of the invention, the wafermodel parameters (T, R, Exp) are solved as described in Eq. (1) or (2).Instead of solving the wafer model parameters (T, R, Exp) according to apredetermined recipe based on the measured positions and based onpredetermined weighing coefficients, the weighing coefficients are alsosolved together with the wafer model parameters (T, R, Exp), with theboundary condition that the sum of the weighing coefficients equals one(i.e. the sum of the alpha's is equal to one).

These 2N equations are solved simultaneously. In matrix notation thefull set of equations (X and Y) may be written as: $\begin{matrix}{{\begin{pmatrix}1 & 0 & {x_{nom}\left( X_{1} \right)} & {- {y_{nom}\left( X_{1} \right)}} \\1 & 0 & {x_{nom}\left( X_{2} \right)} & {- {y_{nom}\left( X_{2} \right)}} \\\vdots & \vdots & \vdots & \vdots \\1 & 0 & {x_{nom}\left( X_{N} \right)} & {- {y_{nom}\left( X_{N} \right)}} \\0 & 1 & {y_{nom}\left( Y_{1} \right)} & {x_{nom}\left( Y_{1} \right)} \\0 & 1 & {y_{nom}\left( Y_{2} \right)} & {x_{nom}\left( Y_{2} \right)} \\\vdots & \vdots & \vdots & \vdots \\0 & 1 & {y_{nom}\left( Y_{N} \right)} & {x_{nom}\left( Y_{N} \right)}\end{pmatrix}\begin{pmatrix}T_{x} \\T_{y} \\{Exp} \\R\end{pmatrix}} = \begin{pmatrix}{{x_{meas}\left( X_{1} \right)} - {x_{nom}\left( X_{1} \right)}} \\{{x_{meas}\left( X_{2} \right)} - {x_{nom}\left( X_{2} \right)}} \\\vdots \\{{x_{meas}\left( X_{N} \right)} - {x_{nom}\left( X_{N} \right)}} \\{{y_{meas}\left( Y_{1} \right)} - {y_{nom}\left( Y_{1} \right)}} \\{{y_{meas}\left( Y_{2} \right)} - {y_{nom}\left( Y_{2} \right)}} \\\vdots \\{{y_{meas}\left( Y_{N} \right)} - {y_{nom}\left( Y_{N} \right)}}\end{pmatrix}} & (3)\end{matrix}$or Ax=b in short. As will be known to a person skilled in the art, theleast on may be expressed as:x =(A ^(T) A)⁻¹ A ^(T) b .

In an exemplary (and non-limiting) application of such a method, thealignment beam (AB) comprises 2 wavelengths or colors (for instance: redand green) and each alignment mark 10 generates seven diffraction linesfor each color. In total, 14 diffraction lines are generated. In thiscase, it may be desirable to determine not only the 4 wafer modelparameters (T_(x), T_(y), R and Exp), but also 14 weighing coefficientsα_(i). Now we introduce a measured position that is a linear combinationof the individual measured order positions: $\begin{matrix}{x_{meas} = {\sum\limits_{i = 1}^{14}\quad{\alpha_{i}x_{{meas},i}}}} & (5)\end{matrix}$for i=1 (the first red measured position) to 14 (the seventh greenmeasured position). In fact, the coefficients α_(i) make up the (static)alignment recipe. Substituting Eq. (5) in Eq. (1), we obtain:${T_{x} + {{Exp}\quad x_{nom}} - {Ry}_{nom}} = {{\sum\limits_{i = 1}^{14}\quad{\alpha_{i}x_{{meas},i}}} - x_{nom}}$which may be rewritten as $\begin{matrix}{{T_{x} + {{Exp}\quad x_{nom}} - {Ry}_{nom} - {\sum\limits_{i = 1}^{14}\quad{\alpha_{i}x_{{meas},i}}}} = {- {x_{nom}.}}} & (6)\end{matrix}$

The above may be written in matrix notation (for one mark) as follows:$\begin{matrix}{{\begin{pmatrix}1 & x_{nom} & {- y_{nom}} & {- x_{{meas},1}} & \cdots & {- x_{{meas},14}}\end{pmatrix}\begin{pmatrix}T_{x} \\{Exp} \\R \\\alpha_{1} \\\vdots \\\alpha_{14}\end{pmatrix}} = \left( {- x_{nom}} \right)} & (6.1)\end{matrix}$

As was stated above, a different set of weighing coefficients may bedetermined for the x and the y direction. In this example it is assumedthat the weighing coefficients α for the diffraction orders in the xdirection are similar to the weighing coefficients β for the diffractionorders in the y direction, so∀i:α_(i)=β_(i)

Of course, it is possible to go through this recipe without imposingthis constraint. Now adding the marks that provide information about they position of the substrate and imposing that the same recipe, i.e. thesame coefficients are applied for the y direction (α_(i)=β_(i)):$\begin{matrix}{{\begin{pmatrix}1 & 0 & x_{nom} & {- y_{nom}} & {- x_{{meas},1}} & \cdots & {- x_{{meas},14}} \\0 & 1 & y_{nom} & x_{nom} & {- y_{{meas},1}} & \cdots & {- y_{{meas},14}}\end{pmatrix}\begin{pmatrix}T_{x} \\T_{y} \\{Exp} \\R \\\alpha_{1} \\\vdots \\\alpha_{14}\end{pmatrix}} = \begin{pmatrix}{- x_{nom}} \\{- y_{nom}}\end{pmatrix}} & (7)\end{matrix}$

The unknowns consist of the four wafer model parameters (T_(x), T_(y),M, R) and the 14 coefficients α_(i). Note, however, that negativecoefficients are also allowed (e.g. for a predictive recipe). The onlyrestriction for the weighing coefficients is that the sum of these mustequal 1, so the number of unknown α_(i)'s in this case is reduced to 13:$\begin{matrix}{{\sum\limits_{i = 1}^{14}\quad\alpha_{i}} = {\left. 1\rightarrow\alpha_{14} \right. = {1 - {\sum\limits_{i = 1}^{13}\quad\alpha_{i}}}}} & (8)\end{matrix}$

Inserting this constraint into equation (6), we obtain $\begin{matrix}{{T_{x} + {Expx}_{nom} - {Ry}_{nom} - {\sum\limits_{i = 1}^{14}{\alpha_{i}x_{{meas},i}}}} = {T_{x} + {Expx}_{nom} - {Ry}_{nom} -}} \\{{\sum\limits_{i = 1}^{13}{\alpha_{i}\left( {x_{{meas},i} - x_{{meas},14}} \right)}} - x_{{meas},14}} \\{= {- x_{nom}}}\end{matrix}$which may be rewritten as $\begin{matrix}{{T_{x} + {Expx}_{nom} - {Ry}_{nom} - {\sum\limits_{i = 1}^{13}{\alpha_{i}\left( {x_{{meas},i} - x_{{meas},14}} \right)}}} = {x_{{meas},14} - {x_{nom}.}}} & (9)\end{matrix}$

The full matrix (7) can now be simplified into: $\begin{matrix}{{\begin{pmatrix}1 & 0 & x_{nom} & {- y_{nom}} & {x_{{meas},14} - x_{{meas},1}} & \cdots & {x_{{meas},14} - x_{{meas},13}} \\\quad & \quad & \quad & \quad & \quad & \quad & \quad \\0 & 1 & y_{nom} & x_{nom} & {y_{{meas},14} - y_{{meas},1}} & \cdots & {y_{{meas},14} - y_{{meas},13}} \\\quad & \quad & \quad & \quad & \quad & \quad & \quad\end{pmatrix}\begin{pmatrix}T_{x} \\T_{y} \\{Exp} \\R \\\alpha_{1} \\\vdots \\\alpha_{13}\end{pmatrix}} = {\begin{pmatrix}{x_{{meas},14} - x_{nom}} \\\quad \\{y_{{meas},14} - y_{nom}} \\\quad\end{pmatrix}.}} & (10)\end{matrix}$

The total number of unknowns is 4 (for T_(x), T_(y), Exp and R)+13 (forα₁ . . . α₁₃)=17. Hence, it may be desirable to measure at least 9 X and8 Y marks (or 9 X and 8 Y marks) on a wafer in order to establish aleast-squares solution of such a system. Solving the wafer model using aleast-squares method may include, e.g., minimizing the followingrelation: $\begin{matrix}{\sum\limits_{\forall i}{\sqrt{\left( {x_{{meas},i} - x_{nom}} \right)^{2} + \left( {y_{{meas},i} - y_{nom}} \right)^{2}}.}} & (11)\end{matrix}$

If only the red (or only the green) color is considered, the number ofunknown parameters reduces to 4 (for T_(x), T_(y), Exp and R)+6 (for α₁. . . α₆)=10, and 5 XY pairs may be sufficient. If only the odddiffraction orders of each color are known, the number of unknownsreduces to 4 (for T_(x), T_(y), Exp and R)+7 (for α₁, α₃, α₅, α₇, α₈,α₁₀, α₁₂)=11, and 5 X and 6 Y marks (or 5 Y and 6 X marks) may besufficient. If both only odd orders and only a single color isconsidered, the number of unknowns reduces to 4 (for T_(x), T_(y), Expand R)+3 (for α₁, α₃, α₅)=7, and 4 X and 3 Y marks (or 4 Y and 3 Xmarks) may be sufficient.

It will be understood that such a method of determining the measuredposition of a substrate W may also be applied to other objects that areprovided with alignment marks 10 capable of generating more than onepositional indicator for a certain direction, such as diffraction orderswhen illuminated by an alignment beam AB. Such a method may, forinstance, also be used to determine the position and orientation ofpatterning means (mask) MA.

A method as described above may be applied to every single substrate Wthat is positioned under the exposure tool. However, one may also decideto determine the optimal weighing coefficients α_(i) for only onesubstrate W and use the results for subsequent substrates W that havebeen subjected to similar process steps. For example, it may beappropriate to assume that such subsequent substrates W will exhibitsimilar features, such that a reasonable calculation of the translation,expansion and/or rotation will be obtained using the same weighingcoefficients. It is also possible to determine the optimal weighingcoefficients α_(i) based on a batch, and use the results for other (e.g.subsequent) batches. Preferably, the weighing coefficients α_(i) aredetermined for every new batch. The first substrate W of a batch is thenused to calculate new weighing coefficients α_(i), that are used forthat batch. In such a case, the dynamic recipe as presented here onlyneeds to be applied for the first substrate W of a batch. The rest ofthe substrates W may than be processed using a static recipe, e.g. usingthe weighing coefficients as calculated as a set of fixed weighingcoefficients.

Also other strategies can be conceived. For instance, it is alsopossible to determine the weighing coefficients α_(i) based on a numberof substrates W. One may also decide to determine the weighingcoefficients α_(i), based on a moving average, for instance, based onthe last 20 substrates W processed. Alternatively, a batch method asdescribed above may be modified by calculating the weighing coefficientsbased on measurements from sample substrates from more than one batch.It will be clear to a person skilled in the art that several strategiesmay be conceived, without departing from the scope of the invention.

For example, one could first run a calibration batch to determineoptimal values for one or more weighing coefficients α_(i), and applythose values to following batches. One could also monitor thecoefficients in time. Variation of a weighing coefficient in time, i.e.a coefficient getting higher or lower, may be a sign that the process isdrifting.

Also, the significance of particular weighing coefficients may bedetermined. If the value of a weighing coefficient has no significantmeaning, for example, the weighing coefficient could be set to zero. Thecalculation may be repeated without the corresponding orders (e.g. theweighing coefficient equals zero). This process could be repeated in aniterative way, e.g., until a significant set of weighing coefficients isdetermined.

Application to so-called multigrating marks is also possible. Amultigrating mark may have, for example, 3 good signals (orders) foreach color. By monitoring the six coefficients, the best signal(grating) can be automatically identified and selected.

The weighing coefficient may further be made dependent on the strengthof the received signal. For example, the strength of the signaloriginating from different diffraction orders may be measured, and if itis too low with respect to a certain threshold, a value of the weighingcoefficient belonging to that diffraction order may be modified (e.g.set to zero).

Whilst specific embodiments of the invention have been described above,it will be appreciated that the invention as claimed may be practicedotherwise than as described. It is explicitly noted that the descriptionof these embodiments is not intended to limit the invention as claimed.

The following table includes a list of symbols as applied herein: T_(x);T_(y) Translation in x and y direction R_(x); R_(y) Rotation of thex-axis and y-axis respectively around the z-axis. Exp_(x); Exp_(y) Waferexpansion in x and y direction x_(nom); y_(nom) Nominal position of amark on the wafer x_(meas,i) and Measured position of a mark on thewafer for, e.g., color y_(meas,i) and order corresponding to i α_(i),β_(i) Recipe weight coefficient in x and y direction for, e.g.,color/order combination corresponding to i

1. A method of determining at least one parameter of a model, the modelproviding information about a position of an object, the object beingprovided with a plurality of alignment marks of which desired positionsare known, the method comprising: measuring a plurality of positionalparameters for each of the plurality of alignment marks, determining theat least one parameter of the model, based on the measured plurality ofpositional parameters, wherein the plurality of positional parametersfor each of the plurality of alignment marks are weighted with weighingcoefficients, and wherein said determining the at least one parameter ofthe model includes determining the numerical value of at least one ofthe weighing coefficients together with the at least one parameter ofthe model.
 2. The method according to claim 1, wherein the at least oneparameter of the model includes at least one of translation, rotation,and expansion.
 3. The method according to claim 1, wherein the pluralityof alignment marks are formed as diffractive elements, and wherein theplurality of positional parameters are determined based on diffractionlines generated by projecting an alignment beam to at least one of theplurality of alignment marks.
 4. The method according to claim 1,wherein the plurality of alignment marks includes at least onemultigrating.
 5. The method according to claim 1, wherein saiddetermining the at least one parameter of the model of the objectincludes determining a least-squares solution of an expression based ondesired and measured positions of at least one of the plurality ofalignment marks.
 6. The method according to claim 1, wherein the atleast one parameter of the model is solved by minimizing the expression$\sum\limits_{\forall i}\sqrt{\left( {x_{{meas},i} - x_{nom}} \right)^{2} + \left( {y_{{meas},i} - y_{nom}} \right)^{2}}$for the plurality of alignment marks, where X_(meas,i) and Y_(meas,i)denote measured positions of the alignment marks based on an i-thpositional parameter in an x-direction and y-direction, respectively;and x_(nom) and y_(nom) denote desired positions in the x- andy-direction, respectively.
 7. The method according to claim 1, whereinsaid determining the at least one parameter of the model of the objectincludes setting the value of a weighing coefficient to zero when thesignal strength of a corresponding positional parameter is below acertain threshold.
 8. The method according to claim 1, wherein theobject is a substrate.
 9. The method according to claim 1, said methodfurther comprising: subsequent to said determining the at least oneparameter of the model of the object, projecting a patterned beam ofradiation onto a target portion of the object.
 10. The method accordingto claim 9, wherein said determining the numerical value of eachweighing coefficient is based on measurements of at least one object,and wherein the method further comprises using the determined numericalvalue of at least one of the weighing coefficients during determinationof the at least one parameter of the model for at least one otherobject.
 11. A lithographic apparatus comprising: a projection systemconfigured to project a patterned beam of radiation onto a targetportion of a substrate, a sensor; a processing unit arranged tocommunicate with the sensor; and a beam generator arranged to project analignment beam to at least one of a plurality of alignment marks, ofwhich desired positions are known, wherein the sensor is arranged tomeasure positional parameters for each of the plurality of alignmentmarks based on the projected alignment beam and to transfer the measuredpositional parameters to the processing unit, and wherein the processingunit is arranged to determine at least one parameter of a modelproviding information about a position of the substrate, based on themeasured positional parameters, and wherein the measured positionalparameters are weighted with weighing coefficients, and wherein theprocessing unit is arranged to determine a numerical value of at leastone of the weighing coefficients together with the at least oneparameter of the model.
 12. The lithographic apparatus according toclaim 11, wherein the at least one parameter of the model includes atleast one of translation, rotation, and expansion.
 13. The lithographicapparatus according to claim 11, wherein the plurality of alignmentmarks are formed as diffractive elements.
 14. The lithographic apparatusaccording to claim 11, wherein the processing unit is arranged todetermine the at least one parameter of the model based on aleast-squares solution of an expression based on desired and measuredpositions of at least one of the plurality of alignment marks.
 15. Thelithographic apparatus according to claim 11, wherein the processingunit is arranged to set the value of a weighing coefficient to zero whenthe signal strength of a corresponding positional parameter is below acertain threshold.
 16. The lithographic apparatus according to claim 11,wherein the apparatus is further configured to project a patterned beamof radiation onto a target portion of the substrate based on the atleast one parameter of the model.
 17. The lithographic apparatusaccording to claim 11, wherein said processing unit is arranged todetermine the numerical value of at least one of the weighingcoefficients based on measurements of at least one substrate, andwherein the processing unit is arranged to use the determined numericalvalue of at least one of the weighing coefficients during adetermination of the at least one parameter of the model for at leastone other substrate.
 18. A method of determining a value for a parameterof a model, the model providing information about a position of anobject, the method comprising: obtaining a plurality of measuredpositions for each of a plurality of alignment marks on the object,solving an expression to obtain the value for the parameter of themodel, the expression being based on (1) the plurality of measuredpositions, (2) at least one desired position for each of the pluralityof alignment marks, and (3) a plurality of weighting coefficients, eachweighting coefficient corresponding to at least one of the plurality ofmeasured positions, wherein a value of at least one of the weightingcoefficients is based on the plurality of measured positions and the atleast one desired position for each of the plurality of alignment marks.19. The method according to claim 18, wherein solving the expressionincludes calculating a least-squares solution of the expression.
 20. Themethod according to claim 18, wherein the value of at least one of theweighting coefficients is based on the value for the parameter of themodel.
 21. The method according to claim 18, wherein the parameter ofthe model includes at least one of translation, rotation, and expansion.22. The method according to claim 18, wherein obtaining a plurality ofmeasured positions for each of a plurality of alignment marks includesdirecting a beam at an alignment mark and measuring at least a portionof a diffraction pattern.